Remark 10.69.7 (Other types of regular sequences). In the paper [Kabele] the author discusses two more regularity conditions for sequences $x_1, \ldots , x_ r$ of elements of a ring $R$. Namely, we say the sequence is Koszul-regular if $H_ i(K_{\bullet }(R, x_{\bullet })) = 0$ for $i \geq 1$ where $K_{\bullet }(R, x_{\bullet })$ is the Koszul complex. The sequence is called $H_1$-regular if $H_1(K_{\bullet }(R, x_{\bullet })) = 0$. One has the implications regular $\Rightarrow$ Koszul-regular $\Rightarrow$ $H_1$-regular $\Rightarrow$ quasi-regular. By examples the author shows that these implications cannot be reversed in general even if $R$ is a (non-Noetherian) local ring and the sequence generates the maximal ideal of $R$. We introduce these notions in more detail in More on Algebra, Section 15.30.

Comment #7023 by Jonathan on

In this remark, the implications regular $\implies$ Koszul-regular $\implies$ $H_1$-regular $\implies$ quasi-regular assume that $R$ is local, while in https://stacks.math.columbia.edu/tag/062D below the first definition (concretely, https://stacks.math.columbia.edu/tag/062F, https://stacks.math.columbia.edu/tag/0CEM, https://stacks.math.columbia.edu/tag/062I) this assumption is not made. Can the assumption that $R$ is local be removed here?

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